We require that x be real, for which the right hand side should be defined. This requirement will place a restriction on the values that f(x) can take. For all the allowed values of f(x), there will exist a valid value of x (a pre-image’ l, or an ‘inverse’). For unallowed values of f(x), there will exist no pre-image or no inverse. Hence f(x) will not take on such values, or in other words, such values will not lie in the range of f. In the expression above, we see that f(x) \( \ne \)1.

These values should form our range. But we have to be a little careful here. y \(\ne \) 1 arises because the quadratic formula gives (1 – y) in the denominator. Suppose that in the (*) equation itself, we put y = 1, reducing the equation to a linear one:

To evaluate the range of such (rational) expressions whose range is not obtainable from simple rearrangement (like in a quadratic expression), we put the expression (in x) equal to some variable y, write x as a function of y (x = g(y)) and find the domain of g; this domain consists of values that y can take or our required range (An example of this sort is described in the unit on functions).